Simplify the following expression: $y = \dfrac{4x^2- 3x- 10}{4x + 5}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(4)}{(-10)} &=& -40 \\ {a} + {b} &=& &=& {-3} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-40$ and add them together. Remember, since $-40$ is negative, one of the factors must be negative. The factors that add up to ${-3}$ will be your ${a}$ and ${b}$ When ${a}$ is ${5}$ and ${b}$ is ${-8}$ $ \begin{eqnarray} {ab} &=& ({5})({-8}) &=& -40 \\ {a} + {b} &=& {5} + {-8} &=& -3 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({4}x^2 +{5}x) + ({-8}x {-10}) $ Factor out the common factors: $ x(4x + 5) - 2(4x + 5)$ Now factor out $(4x + 5)$ $ (4x + 5)(x - 2)$ The original expression can therefore be written: $ \dfrac{(4x + 5)(x - 2)}{4x + 5}$ We are dividing by $4x + 5$ , so $4x + 5 \neq 0$ Therefore, $x \neq -\frac{5}{4}$ This leaves us with $x - 2; x \neq -\frac{5}{4}$.